Evolutionary Games on Complex Networks

Classical Games

The Ultimatum Game

Public Goods

Games

THM:

The more we approach the real social structure the more chances for cooperation to survive!!

• Scale-Free,
• Clustering,
• Multiplexity,
• Adaptivity of growth,
• Group structure

......

Evolutionary Games & Networks

Games in Networks

Social Networks

Possible ways out for the survival

of cooperation

• Kin and Group selection
• Direct and Indirect Reciprocity
• Punishment of defectors

N-person Games

... and the structure of populations!!

Continuous Strategies

Lieberman, Hauert & Nowak,

Nature 433, 312 (2005)

The ultimatum scenario

• Two players are offered some quantity of money to share
• At random, one is assigned the role of proposer and the other that of responder
• The proposer makes an offer (a % of the money) to the responder
• The responder accepts or declines the offer:

Ultimatum experiments

• Most of proposals are about the 40-50% (they are often accepted)
• Most of the proposals below 20% are rejected

• If accepted: the money is split as proposed
• If declined: no one gets anything

Irrationality all over the world

Not Rational!!!

Since 2006...

Complex Networks

Scale-free property

The rational choice is:

• As proposer: offer the smallest

possible amount

• As responder: accept any offer

WHY??

Güth, Schmittberger, Schwarze. An Experimental Analysis of Ultimatum Bargaining.

Journal of Economic Behavior and Organization 3, 367–388 (1982).

How do the social interaction patterns affect cooperative behavior?

Irrationality all over the world

Gender differences

Summary of experiments

Lets model...

• Offers of 40-60% are tolerated
• Small offers (less than 30%) are personal offenses

• Low offers are accepted when roles are assigned at random

• Poorer offers
• Almost no rejection

• A population of N agents.
• Each agent i has the strategy:
• Each agents plays the UG twice: one as proposer ( ) and the other one as responder ( ) with its neighbors ( ).
• The payoff collected by i when playing with j is:

• More Fair offers
• Frequent Rejections

Rejection of small offers are explained as a costly punishment

• Men offers are smaller
• Women accept poor offers

Punishment Cooperation

• So the total payoff of agent i is:

Accumulate payoffs

Set Initial Strategies

Assign Network Topology

(A) Empathetic agents (p=q)

C.F. Camerer, Behavioral Game Theory,

Princeton University Press, 2003

• Then update strategies ...

Assumption: Well mixed (all-to-all) population & a distribution D(p) of offers.

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Deals Diagram

• Cooperation (red)
• Defection (blue)

• Given an agent i with

Empathetic

+S

• The payoff of i given x is given by:

Pragmatic

+2S

Lets simplify a little bit

where

T

• Erdös-Rènyi
• Scale-Free
• Watts-Strogatz
• Modular networks
• ....

Basic Ingredients:

• Set of Strategies
• Payoff Matrix

+2S

2 possible simplifications:

... as usual

Tragedy of the commons!!

CASE A:

Empathetic players

• What accounts for the success of a strategy x?

Remind Replicator Equation: If positive strategy x will be replicated (imitated), otherwise it will be replaced by another x' with better performance ( )

Empathetic Always a partial deal

Only the deal containing the highest offer is closed

Pragmatic Complete deal or nothing

The two deals are closed provided the sum of offers

is larger than 1, otherwise no deal

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Learning rules

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Poisson vs Scale-free Networks:

(A) Empathetic agents (p=q)

Update Strategies

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Measurements

• Start the game again with the new strategies

Does it happen in networks?

• Initially we assign x randomly, then:

D(p) is homogeneous and <P>=0.5

4

• Then:

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

N=10

ER

Brief analysis of pragmatic agents (B)

SF

Payoff only when:

Two frameworks:

• Fixed Cost per Game

Total contribution of a

Cooperator: c(k+1)

• Fixed Cost per Individual

Contribution of a Cooperator

in each group: c/(k+1)

• Initially, the values of x start to decrease reaching a state in which most of the offers are x<0.5

• In a second stage, those individuals offering large (below 0.5) offers are the fittest and thus finally the offers are concentrated around x=0.5.

• Average fraction of cooperators
• Correlations between structural properties and strategies of nodes
• Dynamical behavior of nodes
• Differences between network types

SF networks allow strategies with p>0.5 and small values of p due to hubs

Generous hub:

ER

Two stage dynamics: (i) Deletion of high offers

(ii) Concentration around x=0.5

Rationality does not appear (as in real experiments)

SF

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Social network is SF but...

Scale-free interactions enhance cooperation!!

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

What if offers and rejections are not related?

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Social Punishment

Santos, Santos & Pacheco, Nature 08

Independent p's & q's

Social Punishment

What is the structure of real groups?

Independent p & q (ER networks)

Novelty: Instead of updating single strategies we implement a kind of social punishment:

• After playing the UG the agent with the smallest payoff & its neighbors are selected.

• They are assigned a new strategy (p,q) randomly.

0

Social punishment drives the population to the

Pragmatic scenario:

Social Punishment affects neighbors regardless their benefits

Individuals have to take care of their benefit & of that of their neighbors

Summary

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

• Offers become smaller (30-40%)
• Some agents with q=0 appear
• Rejection reaches its maximum at 30%
• All the offers of more of 40% are accepted
• Common situation p>q

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Gomez-Gardenes et al., CHAOS 11 & EPL 11

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Social Punishment

Social punishment sets an altruistic trend ( ) as

the connectivity of the agent increases

Gomez-Gardenes et al., CHAOS 11 & EPL 11

Cooperation is further enhanced due to group structure

Sinatra, Iranzo, Gomez-Gardeñes, Floría, Latora & Moreno, JSTAT (2009)

Gomez-Gardenes et al., CHAOS 11 & EPL 11

Social Dilemmas on CN's

Cooperation: From Regular to Heterogenous graphs

Erdös-Rènyi

Evolutionary Dynamics

Santos, Pacheco & Lenaerts, PNAS 06

Analysis of the dynamical patterns

Evolutionary

Dynamics

Santos, Pacheco & Lenaerts, PNAS 06

Pure C:

C

Evolution of the partition

C

Weak PD Game

C

Pure D:

D

Distribution across degree classes:

D

We can measure the organization of each class of into clusters

Fluct:

C

Pure Cooperators

D

Scale-free provides an Eden for cooperation

D

2.0

C

Fluctuatings

D

C

1.0

1.5

log(k)

0.8

1.0

time

Gomez-Gardeñes, Campillo, Floría & Moreno, PRL 07

0.6

0.5

0.4

0.2

1.0

Gomez-Gardeñes, Campillo, Floría & Moreno, PRL 07

Explaining the robustness of this Eden

2.0

3.0

1.0

0

2.0

b

The bipolar model

3.0

b

Gomez-Gardeñes, Poncela, Moreno & Floría, J. Theor. Biol. 08

Explaining the robustness of this Eden

Gomez-Gardeñes, Poncela, Moreno & Floría, J. Theor. Biol. 08

Payoff of a Coop. in F:

Effects of Clustering:

Payoff of a Def. in F:

Social

Dilemmas

Effects of Clustering:

Payoff of node 1:

Is this configuration stable for large enough values of b?

Payoff of node 2:

Sufficient Condition:

Gomez-Gardeñes, Poncela, Moreno & Floría, J. Theor. Biol. 08

Gomez-Gardeñes, Poncela, Moreno & Floría, J. Theor. Biol. 08

Thank you for your cooperation!!

Mobility of agents (time-varying interactions)

Cooperation is essential for:

Assenza, Gomez-Gardeñes & Latora, PRE 08

• Major transitions in evolution (multicellular)
• Group defense in animals
• Social welfare
• Global sustainability

Agents move in a 2D plane with constant velocity v

Mobility of agents (time-varying interactions)

Assenza, Gomez-Gardeñes & Latora, PRE 08

At each time step t :

• They set contacts with their nearest nodes (up to some distance R)
• A Random Geometric Graph is formed
• They play a PD game
• They update strategies
• Continue moving and start again at time t+1

Adaptive networks

Velocity is highly detrimental for cooperation

Adaptive networks

• Network structure evolves as a function of the evolutionary game (strategies, payoffs, ...)
• The network can be already grown (rewiring) or be growing coupled to the game dynamics.

What is the role of the velocity v ?

Hierarchical clustering

There is an abrupt transition when the velocity is the control parameter

Cooperation is Enhanced!

Evolutionary Preferential Attachment:

S. Meloni et al. PRE 79, 067101 (2009)

Games in Interdependent networks

Poncela, Gomez-Gardeñes, Sanchez, Floría & Moreno, PLoS One 08

Between agents of the same population:

Games in Interdependent networks

C

D

Poncela, Gomez-Gardeñes, Sanchez, Floría & Moreno, PLoS One 08

C

: fraction of Coops. in population 1

Coupled Replicator eqs.

D

PD game

Games in Interdependent networks

: fraction of Coops. in population 2

Between agents of different populations:

C

D

ER network <k>=6

C

D

HD game

Gomez-Gardeñes, Gracia-Lázaro, Floría & Moreno, PRE 86, 056113 (2012)

Effects due to multiplexity (coupled networks)

• Individuals act simultaneously in "m" networks.
• They can take different strategies in each of them.
• Individuals only have access to the total payoffs of others.

Gomez-Gardeñes, Reinares, Arenas & Floría, Nature Sci. Rep. 12

Jesús Gómez-Gardeñes, GOTHAM lab,

Insitute for Biocomputation and Physics of Complex Systems. Universidad de Zaragoza